Discrete Single-Photon Quantum Walks with Tunable Decoherence

Quantum walks have a host of applications, ranging from quantum computing to the simulation of biological systems. We present an intrinsically stable, deterministic implementation of discrete quantum walks with single photons in space. The number of optical elements required scales linearly with the number of steps. We measure walks with up to 6 steps and explore the quantum-to-classical transition by introducing tunable decoherence. Finally, we also investigate the effect of absorbing boundaries and show that decoherence significantly affects the probability of absorption.

Figure 1

Experimental schematic. (a) Photons created via spontaneous parametric down-conversion in a PPKTP crystal are injected into free-space mode 0. Arbitrary initial coin (polarization) states are prepared by the first polarizing beam splitter (PBS) and wave plate combination. Six pairs of coin ($C_i$) and shift ($S_i$) operators implement six steps of the walk. Coincident detection of photons at detector D2 and D1 (4.4 ns time window) herald a successful run of the walk. (b) Details of our optical mode numbering convention for the first two steps. The dashed lines trace out one of the interferometers which are used to align the quantum walk. (c) A relative angle between two beam displacers reduces the recombined photon’s temporal ($\Delta t$) and spatial ($\Delta x$) mode overlap, thereby implementing tunable decoherence.

Figure 2

Probability distributions for successive steps of the (a) quantum and (b) fully decohered (classical) walks up to the sixth step. Dashed lines show experimental data and solid lines show theoretical predictions. Probabilities are obtained by normalizing photon counts at each position to the total number of counts for the respective step. The insets show horizontal scans across the walk lattice for the five-step quantum walk (coupled into single-mode fiber) and decohered random walk (multimode fiber), respectively. (c) Normalized standard deviation of the probability distribution for quantum (black circles) and classical (red triangles) walks for 1 to 6 steps. Lines show the theoretical values; error bars are smaller than symbol size.

Figure 3

Transition of a 5-step quantum walk to a classical random walk. The decoherence parameter $q$ for a given data set was obtained from a least-squares fit to theory, Eq. (2). All error bars are smaller than symbol size; the shaded area represents the theoretical probability distributions.

Figure 4

Transmission probability of quantum (black circles) and classical (red triangles) walkers with an absorber located at position $-1$ and initial coin state $|L⟩$. The transmission was obtained as the ratio of the number of transmitted photons measured with absorbers in place to the number measured without them. Error bars are smaller than symbol size. The insets show the fifth step walk with an absorber at position $-1$ (red columns) compared to the original walk (blue columns) for the quantum (QW) and classical case (CW).